Ideas
Substitution
After building an algebraic expression we can solve it by substitution, where we replace pronumerals with numeric values.
Consider the following table of values:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 4 | 7 | 10 | 13 | 16 |
We can construct an equation describing the relationship between x and y:y=3x+1
What is the next number in the pattern?
We can solve this problem by using substitution. A key part of substitution is understanding the equation and identifying which pronumeral to substitute.
In this table of values, the x-values represent the position in the pattern. For example, the x=1 column represents the 1st position, the x=4 column represents the 4th value in the pattern, and so on. The y-values represent the numbers in the pattern, so the 1st value is 4 and the 4th value is 13.
We are trying to find the next value, which in this case is the 6th value. In other words, we want to find what the value of y is when the value of x is 6.
In an algebraic expression, the term 3x means 3\times x. So if we substituted x=4 into the equation then the term is equal to 3\times 4= 12 and not 34.
Let's now perform the substitution, using x=6:
| \displaystyle y | \displaystyle = | \displaystyle 3x+1 | |
| \displaystyle = | \displaystyle 3\times 6+1 | Substitute x=6 into the equation | |
| \displaystyle = | \displaystyle 18+1 | Simplify the product | |
| \displaystyle = | \displaystyle 19 | Evaluate |
We can see that the 6th number in the pattern is 19.
Now we could have found this value by adding 3 to the 5th number, since the numbers in the pattern go up by 3 each step. But what if we are asked to find the 20th (or the 105\text{th}) number in the pattern? We don't want to add 3 twenty (or one hundred and five) times.
Substitution allows us to find the answer directly, no matter what number we choose. We can find the 20th number in the pattern (x=20):
| \displaystyle y | \displaystyle = | \displaystyle 3x-5 | Write the equation |
| \displaystyle = | \displaystyle 3\times 20+1 | Substitute x=20 into the equation | |
| \displaystyle = | \displaystyle 60+1 | Simplify the product | |
| \displaystyle = | \displaystyle 61 | Evaluate |
.. and the 105\text{th} number (x=105):
| \displaystyle y | \displaystyle = | \displaystyle 3x-5 | Write the equation |
| \displaystyle = | \displaystyle 3\times 105 -5 | Substitute x=105 into the equation | |
| \displaystyle = | \displaystyle 315+1 | Simplify the product | |
| \displaystyle = | \displaystyle 316 | Evaluate |
Examples
Example 1
Find the value of 9+m when m=3.
Example 2
Find the value of \dfrac{u}{9} when u=54.
Example 3
Evaluate 6x+4y+6 when x=3 and y=5.
Substitution is the replacing of the pronumerals with numbers.
A key part of substitution is understanding the equation and identifying which pronumeral to substitute.